Theoretical design of the large topological magnetoelectric effect in the Co-intercalated NbS$_2$ structure

TL;DR

The paper proposes and validates a platform in which antichiral (staggered) 3$q$ spin textures in a two-layer Co intercalation between NbS$_2$ sheets give rise to a large topological magnetoelectric (axion-like) response, distinct from the large anomalous Hall effect seen with uniform chirality. Using DFT+U, Wannier-based Berry-curvature analysis, and controlled strain, the authors show strain-tunable interlayer coupling can switch between chiral and anti-chiral orders, enabling or suppressing the AHE. They quantify the magnetoelectric coupling $\alpha^{zz}$, finding it can reach up to $\sim 0.9\,e^2/2h$, with the anti-chiral state dominated by the Chern-Simons-like contribution from layer-dependent Berry curvature. This work highlights a route to engineer tunable topological magnetoelectric effects in intercalated transition-metal dichalcogenides via strain and magnetic ordering.

Abstract

A triangular Co-ion lattice intercalated between 1-H NbS$_2$ layers can exhibit a large anomalous Hall effect (AHE) due to the finite scalar spin chirality originating from the non-coplanar $3q$ ordering of Co spins. This large AHE occurs when the scalar spin chirality is uniform in all Co layers, as indeed found in the Co$_{1/3}$NbS$_2$ case [Phys. Rev. Mater. 6, 024201 (2022)]. However, if the spin chirality were staggered with the opposite signs in the adjacent Co layers, the net AHE would disappear, yielding instead the topological magneto-electric effect. Here, we theoretically verify that a transverse electric field generates a finite orbital magnetization under such conditions, consistent with the axion-like coupling. Using first-principles calculations, we show that the resulting magneto-electric coupling, $α^{zz}$ can be as large as 0.9 $e^2/2h$. We also demonstrate that the inter-layer magnetic coupling in these materials can be tuned by strain, enabling the switching between the AHE and the axionic states.

Figures (6)

• Figure 1: (a) The crystal structure of vacuum/(NbS$_2$)$_3$/Co/ (NbS$_2$)$_3$/Co/(NbS$_2$)$_3$/vacuum thin film using the slab geometry, (b) The schematic spin structure of Co ions in the $3q$ anti-chiral state.
• Figure 2: The energy difference ($\Delta E$) of the magnetic states between the FM and AFM inter-layer magnetic couplings for $1q-$AFM (black diamond dots) and $3q-$AFM (red square dots). $1q-$AFM1(2) and $3q-$Chiral(Anti-chiral) states show the AFM(FM) inter-layer coupling between the Co layers. The anti-chiral $3q$ magnetic state is favored in the tensile (positively) strained thin film structure.
• Figure 3: (a) The orbital magnetization $M_{orb}$ per magnetic unit cell for the anti-chiral (AC) magnetic structure under the 2% tensile strain calculated as a function of the orbital energy difference $\delta E$. The total $M_{orb}$ consists of the local circulation ($M^{LC}_{orb}$) and the itinerant circulation ($M^{IC}_{orb}$) terms. (b) The magnetoelectric coupling $\alpha$ computed from the linear slope of $M_{orb}$ vs $\delta E$.
• Figure 4: (a) The orbital magnetic moment per magnetic unit cell compared for the anti-chiral (AC) and the chiral (C) magnetic structures under the 2% tensile strain. (b) The anomalous Hall conductivity $\sigma_{xy}$ computed using the Berry curvature for both structures.
• Figure 5: The decomposition of the magnetoelectric coupling $\alpha^{zz}$ for the anti-chiral magnetic structure into the Chern-Simon term ($\alpha^{CS}$) and the Kubo-like term ($\alpha^{Kubo}$).